Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

TL;DR

This paper establishes a deep connection between orbit and flow equivalence of shift spaces of finite type using groupoid techniques, extending previous results to more general cases and linking these concepts to graph C*-algebras.

Contribution

It generalizes known results by proving equivalences between orbit and flow equivalence for shifts of finite type and relates these to groupoid isomorphisms and algebraic invariants.

Findings

Orbit equivalence implies flow equivalence for shifts of finite type.

Groupoid isomorphisms characterize orbit equivalence of one-sided shifts.

Flow equivalence corresponds to stable isomorphism of associated groupoids and C*-algebras.

Abstract

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph $C^*$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz-Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.